Recursive Relation for Counting the Complexity of Butterfly Map

Communications in Advanced Mathematical Sciences, 2021

In this paper, we obtain closed formulas for the number of reachable vertices in labelled plane trees by paths lengths, sinks, leaf sinks, first children, left most path, non-first children, and non-leaves. Our counting objects are plane trees having their edges oriented from a vertex of lower label towards a vertex of higher label. For each statistic, we obtain the average number of reachable vertices. Moreover, we obtain a counting formula for the number of plane trees on n vertices such that exactly k ≤ n are reachable from the root.

2008

Abstract—The Italian Airport Network (IAN) is considered. The description in term of a mathematical graph is given and its topological properties are approached by means of a new mathematical tool: the multiple addendials. The connection degree and the betweenness centrality distributions in the IAN follow a power-law behaviour, well known in literature like a Double Pareto Law. This leads to the definition of the IAN as a scale-free network.

Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88, 1988

The pow of Butterfly-type networks relative to other Propose multicomputer interconnection networks is studied. by considering how efficiently the Butterfly can simulate the other networks. Simulation is represented formally via graph embeddings, so the topic here becomes: How efficiently can one embed the graph underlying a given network in the graph underlying the Butterfly network? The efficiency of an embedding of a graph G In a graph H Is measured In tefrms of. the dilation, or, the maximum amount that any edge of GIs "stretched" by the embedding; theeiqWanslon, or, the ratio of the number of vertices of H to the number of vertices of G. Three general results about embeddings In Butterfly-type graphs are established here, tha expose a umber of simulations by Butterfly-type networks. which are optimal (to within constant factors): (1) Any plate binary tree can be embedded In a Butterfly graph, with simultaneous dilation 0(1) and nsion 0(1). (2) Any n-vertex graph having a r2~-blfurcator of size S fl(log n) can be embedded In Butterfly graph with simultans dilation O(Iog S) and expansion 0(1). (3) Anyebeddngvf plnrpG In aButterfly graph must havedlation f fog E (G)J/9 (G)): E (G) Is the size of the smallest 1/3-2/3v separator of G; # (G) Is the size of G3's largest interior face. Corollaries Include: (a) The n-vertex X-tree be embedded In the Butterfly with simultaneous dilation Opog log n) and expansion 0(1); no embeddi yields smaller dilaftion Independent of expansion. (b) Every embedding of the n x n mesh In Butterfly has diltion n (log n); any expansion-O(i) embedding of the mesh In the Butterfly achieves this d Thes results. which extend to Butterfly-like graphs such as the Cube-Connected Cycles and Beries networks, supply the first examples of graphs that can be embedded more efficienry In the Hypercube thanIn the Butterfly.

Pattern Recognition and Machine Intelligence, 2011

In this paper we analyze the topological properties of airport network of India (ANI) using graph theoretic approach. We show that such an analysis can be useful not only in planning the infrastructure and growth of the air-traffic connectivity, but also in managing the flow of transportation during emergencies such as accidental failure of the airport, close down of the airport due to unexpected climate changes, terrorist attacks, etc. Knowledge of the connectivity pattern and load on various routes can also help in making judicious decisions for reduction of flights to contain the spread of the infectious disease.

There exist many algorithms for producing the spanning trees of a graph with better time and space complexities. In this research study, we are presenting a study on number of spanning trees and a technique based on the basic cycle to find the number of spanning trees and also the structure of all the spanning trees of a labeled and undirected graph.

2019

The airlines in the real world form small-world network. This implies that they are constructed with an ad hoc strategy. The small-world network is not so bad from the viewpoints of customers and managers. The customers can fly to any destination through a few airline hubs, and the number of airlines is not so many comparing to the number of airports. However, clearly, it is not the best solution in either viewpoint since there is a trade off. In this paper, one of the extreme cases, which is the standpoint of the manager, is considered; we assume that customers are silent and they never complain even if they are required to transit many times. This assumption is appropriate for some transportation service and packet communication. Under this assumption, the airline problem is to construct the least cost connected network for given distribution of the populations of cities with no a priori connection. First, we show an efficient algorithm that produces a good network which is minimi...

2009

Let t(G) denote the number of spanning trees of a graph G. A chain of two connected vertices u,v(dG(u),dG(v) � 3) in G, denoted by Lk, is defined as a path of G and dG(p) = 2 for all p 2 V (Lk) { u,v}, where k is the length of the path. In this paper, we investigate the relationship between t(G) and Lk of a graph G. In particular, the relationship between t(G) and Lk of �-optimal graph G is considered.

The European Physical Journal Special Topics, 2013

Air transport is a key infrastructure of modern societies. In this paper we review some recent approaches to air transport, which make extensive use of theory of complex networks. We discuss possible networks that can be defined for the air transport and we focus our attention to networks of airports connected by flights. We review several papers investigating the topology of these networks and their dynamics for time scales ranging from years to intraday intervals, and consider also the resilience properties of air networks to extreme events. Finally we discuss the results of some recent papers investigating the dynamics on air transport network, with emphasis on passengers traveling in the network and epidemic spreading mediated by air transport.

We study complex networks with weights w ij associated with each link connecting node i and j. The weights are chosen to be correlated with the network topology in the form found in two real world examples: ͑a͒ the worldwide airport network and ͑b͒ the E. Coli metabolic network. Here w ij ϳ x ij ͑k i k j ͒ ␣ , where k i and k j are the degrees of nodes i and j, x ij is a random number, and ␣ represents the strength of the correlations. The case ␣ Ͼ 0 represents correlation between weights and degree, while ␣ Ͻ 0 represents anticorrelation and the case ␣ = 0 reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, ᐉ opt , with the system size N in strong disorder for scale-free networks for different ␣. We find two different universality classes for ᐉ opt in strong disorder depending on ␣: ͑i͒ if ␣ Ͼ 0, then for Ͼ2 the scaling law ᐉ opt ϳ N 1/3 , where is the power-law exponent of the degree distribution of scale-free networks, and ͑ii͒ if ␣ Յ 0, then ᐉ opt ϳ N opt with opt identical to its value for the uncorrelated case ␣ = 0. We calculate the robustness of correlated scale-free networks with different ␣ and find the networks with ␣ Ͻ 0 to be the most robust networks when compared to the other values of ␣. We propose an analytical method to study percolation phenomena on networks with this kind of correlation, and our numerical results suggest that for scale-free networks with ␣ Ͻ 0, the percolation threshold p c is finite for Ͼ3, which belongs to the same universality class as ␣ =0. We compare our simulation results with the real worldwide airport network, and we find good agreement.

IEICE Trans. Inf. Syst., 2021

This paper deals with the problem of enumerating 3-edgeconnected spanning subgraphs of an input plane graph. In 2018, Yamanaka et al. proposed two enumeration algorithms for such a problem. Their algorithm generates each 2-edge-connected spanning subgraph of a given plane graph with n vertices in O(n) time, and another one generates each k-edgeconnected spanning subgraph of a general graph with m edges in O(mT ) time, where T is the running time to check the k-edge connectivity of a graph. This paper focuses on the case of the 3-edge-connectivity in a plane graph. We give an algorithm which generates each 3-edge-connected spanning subgraph of the input plane graph in O(n2) time. This time complexity is the same as the algorithm by Yamanaka et al., but our algorithm is simpler than theirs. key words: enumeration, algorithm, spanning subgraph, edge-connectivity

International journal of mathematics and soft computing, 2012

The all-ones problem is an NP-complete problem introduced by Sutner [11], with wide applications in linear cellular automata. In this paper, we solve the all-ones problem for some of the widely studied architectures like binomial trees, butterfly, and benes networks.

Parallel Computing, 2002

In many parallel applications, the need for broadcasting, scattering, gathering or gossiping is crucial. Many collective communication algorithms have been studied for different topologies of interconnection networks such as hypercubes, meshes, De Bruijn and star graphs. In this paper we study some communication procedures on the binary wrapped butterfly BWBðnÞ of dimension n interconnection networks. We consider the BWBðnÞ as a point-topoint interconnection network. Communication is assumed to be full duplex, all-ports with a linear communication model and is based on store-and-forward techniques. The BWBðnÞ is a constant degree 4 Cayley graph. Vadapalli and Srimani gave a new representation of the BWBðnÞ that bring some convenience in studying the topological properties and fault tolerance. Using this new representation, we propose an improved one-to-all broadcast algorithm, based on a spanning tree of optimal height. We present a technique based on rotative trees for constructing multiple spanning trees that would be used to derive: a fault tolerant one-to-all broadcast, a scattering, a gathering algorithms and theirs fault tolerant version.

The aim of this study is to characterize territorial and topological levels in the structure and evolution of the global air transportation network. Using data from the ITA database for both 2000 and 2004, we analyze the topology of these networks with a new approach based on weighted graphs from a small world perspective (Watts, 1999; Barabasi, 2002; Barat et al.2005). A first step is performed through graph clustering algorithms unfolding the different “small worlds ” together making up the whole network. The clusters are seen as communities and are captured with the use of edge metrics conveying topological properties of the underlying communities. The metric we use is that from Auber, Chiricota et al. 2003 (see also (Amiel, Melançon et al. 2005)) which compares to Burt’s network constraint [(Burt 2000); see also (Burt 2005)]. The outcome of the clustering procedure provides a multilevel presentation of the most interconnected airports in the world. In a second stage, we identify...

International …, 2006

The paper unveils « new reticular territories » implicitly defined in air transport networks. We describe an approach based on weighted graphs (induced from city to city air passenger traffic) from which we infer strongly connected components, themselves organized into sub-networks. The computation can be iterated within each component (or cluster) until each node or city appears as such in a component. The method leads to a multilevel presentation of the most connected cities in the world, underlining the necessary steps to go through when traveling worldwide, described as a path through the different levels and components. These components sometimes appear as being founded on spatial rules, but we also observed specialized subgroups emerging from economical or institutional logics such as partnership strategies. Our methods open new perspectives to study and represent networks or spatial interaction of cities systems at every level of geographical scale, from local to global.

IEEE Transactions on Computers, 1991

In this paper we present embeddings of complete binary trees into butterfly networks with or without wrap-around connections. Let m be an even integer and q = m+ LlogmJ-l. We show how to embed a 2 q +I-I-node complete binary tree T(q) into a (m + 1)2 111 +1_ node wrap-around butterfly B1l1(m +1) with a dilation of 4, and how to embed T(q) into a (m + 2)2 111 + 2 _node wrap-around butterfly B 1l1 (m +2) with an optimal dilation of 2. We also present an embedding of a wrap-around butterfly B 1l1 (m) into a (m + 1)2 m-node no wrap-around butterfly B(m) with a dilation of 3. Using this embedding we show that T(q) can be embedded into a no wrap-around butterfly B(m +1) (resp. B(m + 2) witb a dilation of 8 (resp. 5).